A064894 -id:A064894 - cp's OEIS frontend

A064895 Binary concentration of n. Replace 2^e_k with 2^(e_k/g(n)) in binary expansion of n, where g(n) = GCD of exponents e_k = A064894(n).

0, 1, 2, 3, 2, 3, 6, 7, 2, 3, 10, 11, 12, 13, 14, 15, 2, 3, 18, 19, 6, 7, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 2, 3, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 2, 3, 66, 67, 10, 11, 70, 71, 6, 7, 74, 75

Offset: 0

Marc LeBrun, Oct 11 2001

			577 = 2^0 + 2^6 + 2^9, GCD(0,6,9) = 3, a(577) = 2^(0/3)+2^(6/3)+2^(9/3) = 13.

Rémy Sigrist, Table of n, a(n) for n = 0..8192

Cf. A000079, A064894.

A064895[n_] := With[{e = Flatten[Position[Reverse[IntegerDigits[n, 2]], 1]] - 1}, Total[2^(e/Max[Apply[GCD, e], 1])]];
Array[A064895, 100, 0] (* Paolo Xausa, Feb 13 2024 *)

a(n) = { my (b=vector(hammingweight(n))); for (i=1, #b, n-=2^b[i]=valuation(n,2);); b /= max(1, gcd(b)); sum(i=1, #b, 2^b[i]); } \\ Rémy Sigrist, Oct 16 2022

If n = 2^(g(n)e0) + 2^(g(n)e1) +... then a(n) = 2^e0 + 2^e1 +...

A056538 Irregular triangle read by rows: row n lists the divisors of n in decreasing order.

Original entry on oeis.org

1, 2, 1, 3, 1, 4, 2, 1, 5, 1, 6, 3, 2, 1, 7, 1, 8, 4, 2, 1, 9, 3, 1, 10, 5, 2, 1, 11, 1, 12, 6, 4, 3, 2, 1, 13, 1, 14, 7, 2, 1, 15, 5, 3, 1, 16, 8, 4, 2, 1, 17, 1, 18, 9, 6, 3, 2, 1, 19, 1, 20, 10, 5, 4, 2, 1, 21, 7, 3, 1, 22, 11, 2, 1, 23, 1, 24, 12, 8, 6, 4, 3, 2, 1, 25, 5, 1, 26, 13, 2, 1, 27, 9

Offset: 1

Antti Karttunen, Jun 20 2000

Old name was "Replace n by its divisors in reverse order."
This gives the second elements of the ordered pairs (a,b), a >= 1, b >= 1, ordered by their product ab.
T(n,k) = n / A027750(n,k) = A027750(n,n-k+1), 1 <= k <= A000005(n). - Reinhard Zumkeller, Sep 28 2014
The 2nd column of the triangle is the largest proper divisor (A032742). - Charles Kusniec, Jan 30 2021

			Triangle begins:
1;
2, 1;
3, 1;
4, 2, 1;
5, 1;
6, 3, 2, 1;
7, 1;
8, 4, 2, 1;
9, 3, 1;
10, 5, 2, 1;
11, 1;
12, 6, 4, 3, 2, 1;
13, 1;
14, 7, 2, 1;
15, 5, 3, 1;
16, 8, 4, 2, 1;
17, 1;
18, 9, 6, 3, 2, 1;
19, 1;
20, 10, 5, 4, 2, 1;

Reinhard Zumkeller, Rows n = 1..1000 of triangle, flattened
Omar E. Pol, Illustration of the divisors of n - _Omar E. Pol_, Nov 22 2009

Cf. A027750 for the first elements, A056534, A168017, A000005 (row lengths), A000203 (row sums), A032742 (largest proper divisor).

a056538 n k = a056538_tabf !! (n-1) !! (k-1)
a056538_row n = a056538_tabf !! (n-1)
a056538_tabf = map reverse a027750_tabf
-- Reinhard Zumkeller, Sep 28 2014

Magma
```
[Reverse(Divisors(n)) : n in [1..30]];
```

map(op,[seq(reverse(sort(divisors(j))),j=1..30)]);
cdr := proc(l) if 0 = nops(l) then ([]) else (l[2..nops(l)]): fi: end:
reverse := proc(l) if 0 = nops(l) then ([]) else [op(reverse(cdr(l))), l[1]]; fi: end:

Table[Reverse@ Divisors@ n, {n, 27}] // Flatten (* Michael De Vlieger, Jul 27 2016 *)

row(n)=Vecrev(divisors(n)) \\ Charles R Greathouse IV, Sep 02 2015

a(n) = A064894(A064896(n)).

Definition revised by N. J. A. Sloane, Jul 27 2016

A064896 Numbers of the form (2^(m*r)-1)/(2^r-1) for positive integers m, r.

Original entry on oeis.org

1, 3, 5, 7, 9, 15, 17, 21, 31, 33, 63, 65, 73, 85, 127, 129, 255, 257, 273, 341, 511, 513, 585, 1023, 1025, 1057, 1365, 2047, 2049, 4095, 4097, 4161, 4369, 4681, 5461, 8191, 8193, 16383, 16385, 16513, 21845, 32767, 32769, 33825, 37449, 65535, 65537

Offset: 1

Marc LeBrun, Oct 11 2001

Binary expansion of n consists of single 1's diluted by (possibly empty) equal-sized blocks of 0's.
According to Stolarsky's Theorem 2.1, all numbers in this sequence are sturdy numbers; this sequence is a subsequence of A125121. - T. D. Noe, Jul 21 2008
These are the numbers k > 0 for which k + 2^m = k*2^n + 1 has a solution m,n > 0. For k > 1, these are numbers k such that (k - 2^x)*2^y + 1 = k has a solution in positive integers x,y. In other words, (k - 1)/(k - 2^x) = 2^y for some x,y > 0. If t = (2^m - 1)/(2^n - 1) is a term of this sequence (i.e. if and only if n|m), then t' = t + 2^m = t*2^n + 1 is also a term. Primes in this sequence (A245730) include: all Mersenne primes (A000668), all Fermat primes (A019434), and other primes (73, 262657, 4432676798593, ...). - Thomas Ordowski, Feb 14 2024

			73 is included because it is 1001001 in binary, whose 1's are diluted by blocks of two 0's.

T. D. Noe, Table of n, a(n) for n=1..1000
T. Chinburg and M. Henriksen, Sums of k-th powers in the ring of polynomials with integer coefficients, Acta Arithmetica, 29 (1976), 227-250.
K. B. Stolarsky, Integers whose multiples have anomalous digital frequencies, Acta Arithmetica, 38 (1980), 117-128.

Cf. A064894, A056538.
Cf. A076270 (k=3), A076275 (k=4), A076284 (k=5), A076285 (k=6), A076286 (k=7), A076287 (k=8), A076288 (k=9), A076289 (k=10).
Primes in this sequence: A245730.

f := proc(p) local m,r,t1; t1 := {}; for m from 1 to 10 do for r from 1 to 10 do t1 := {op(t1), (p^(m*r)-1)/(p^r-1)}; od: od: sort(convert(t1,list)); end; f(2); # very crude!
# Alternative:
N:= 10^6: # to get all terms <= N
A:= sort(convert({1,seq(seq((2^(m*r)-1)/(2^r-1),m=2..1/r*ilog2(N*(2^r-1)+1)),r=1..ilog2(N-1))},list)); # Robert Israel, Jun 12 2015

lista(nn) = {v = [1]; x = (2^nn-1); for (m=2, nn, r = 1; while ((y = (2^(m*r)-1)/(2^r-1)) <=x, v = Set(concat(v, y)); r++);); v;} \\ Michel Marcus, Jun 12 2015

A064894(a(n)) = A056538(n).

A272011 Irregular triangle read by rows: strictly decreasing sequences of nonnegative numbers given in lexicographic order.

Original entry on oeis.org

0, 1, 1, 0, 2, 2, 0, 2, 1, 2, 1, 0, 3, 3, 0, 3, 1, 3, 1, 0, 3, 2, 3, 2, 0, 3, 2, 1, 3, 2, 1, 0, 4, 4, 0, 4, 1, 4, 1, 0, 4, 2, 4, 2, 0, 4, 2, 1, 4, 2, 1, 0, 4, 3, 4, 3, 0, 4, 3, 1, 4, 3, 1, 0, 4, 3, 2, 4, 3, 2, 0, 4, 3, 2, 1, 4, 3, 2, 1, 0, 5, 5, 0, 5, 1, 5, 1

Offset: 0

Peter Kagey, Apr 17 2016

Length of n-th row given by A000120(n);
Maximum of n-th row given by A000523(n);
Minimum of n-th row given by A007814(n);
GCD of n-th row given by A064894(n);
Sum of n-th row given by A073642(n + 1).
n-th row begins at index A000788(n - 1) for n > 0.
The first appearance of n is at A001787(n).
a(A001787(n) + 1) = a(A001787(n)) for all n > 0.
a(A001787(n) + 2) = 0 for all n > 0.
a(A001787(n) + 3) = a(A001787(n)) for all n > 1.
a(A001787(n) + 4) = 1 for all n > 1.
a(A001787(n) + 5) = a(A001787(n)) for all n > 1.
Row n < 1024 lists the digits of A262557(n). - M. F. Hasler, Dec 11 2019

			Row n is given by the exponents in the binary expansion of n. For example, row 5 = [2, 0] because 5 = 2^2 + 2^0.
Row 0: []
Row 1: [0]
Row 2: [1]
Row 3: [1, 0]
Row 4: [2]
Row 5: [2, 0]
Row 6: [2, 1]
Row 7: [2, 1, 0]

Peter Kagey, Table of n, a(n) for n = 0..10000

Cf. A000120, A000523, A001787, A007814, A064894, A073642.
Cf. A133457 gives the rows in reverse order.
Cf. A272020, A262557.

Map[Length[#] - Flatten[Position[#, 1]] &, IntegerDigits[Range[50], 2]] (* Paolo Xausa, Feb 13 2024 *)

apply( A272011_row(n)=Vecrev(vecextract([0..exponent(n+!n)],n)), [0..39]) \\ For n < 2^10: row(n)=digits(A262557[n]). There are 2^k rows starting with k, they start at row 2^k. - M. F. Hasler, Dec 11 2019

A271410 LCM of exponents in binary expansion of 2n.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 6, 6, 4, 4, 4, 4, 12, 12, 12, 12, 5, 5, 10, 10, 15, 15, 30, 30, 20, 20, 20, 20, 60, 60, 60, 60, 6, 6, 6, 6, 6, 6, 6, 6, 12, 12, 12, 12, 12, 12, 12, 12, 30, 30, 30, 30, 30, 30, 30, 30, 60, 60, 60, 60, 60, 60, 60, 60, 7, 7, 14, 14, 21, 21, 42

Offset: 0

Peter Kagey, Apr 11 2016

			a(2) = lcm(2) = 2 because 2*2 = 2^2;
a(3) = lcm(1, 2) = 2 because 2*3 = 2^1 + 2^2;
a(7) = lcm(1, 2, 3) = 6 because 2*7 = 2^3 + 2^2 + 2^1.

Peter Kagey, Table of n, a(n) for n = 0..10000

Cf. A029931, A064894, A073642, A096111, A116417.

lcm[n_]:=Module[{idn2=IntegerDigits[n,2]},LCM@@Pick[Reverse[Range[ Length[ idn2]]], idn2,1]]; Join[{1},Array[lcm,100]] (* Harvey P. Dale, Jan 24 2019 *)

a(n) = my(ve = select(x->x==1, Vecrev(binary(2*n)), 1)); lcm(vector(#ve, k, ve[k]-1)); \\ Michel Marcus, Apr 12 2016

a(n)=lcm(Vec(select(x->x, Vecrev(binary(n)), 1))) \\ Charles R Greathouse IV, Apr 12 2016

from math import lcm
def A271410(n): return lcm(*(i for i, b in enumerate(bin(n)[:1:-1],1) if b == '1')) # Chai Wah Wu, Dec 12 2022

cp's OEIS Frontend

A064895 Binary concentration of n. Replace 2^e_k with 2^(e_k/g(n)) in binary expansion of n, where g(n) = GCD of exponents e_k = A064894(n).

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A056538 Irregular triangle read by rows: row n lists the divisors of n in decreasing order.

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A064896 Numbers of the form (2^(m*r)-1)/(2^r-1) for positive integers m, r.

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A272011 Irregular triangle read by rows: strictly decreasing sequences of nonnegative numbers given in lexicographic order.

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A271410 LCM of exponents in binary expansion of 2n.

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